Something I’ve been thinking about is the way we teach students to solve systems of linear equations using substitution. Consider for example the following system of linear equations: y = 3x + 4 and 2y – 3x = 16. Many textbooks would suggest the way forward is to solve one of the equations for a particular value and substitute it in to the other equation. In my example above, students would be encouraged to notice y is already “by itself” so they should substitute 3x + 4 for the y in the second equation to get 2(3x + 4 ) – 3x = 16.
This is not incorrect, but it is also not the only way to solve this system. Students could just as easily solve the first equation for 3x and substitute that in for the 3x in the second equation to get 2y – y – 4 = 16. I think this equation is actually easier to solve than the first. You may be thinking, okay so there are two ways, big woop. Well friends, there are other ways. You could rewrite the second equation so that you can more easily see 3x + 4 like so, 2y = 3x + 4 + 12 and then replace the 3x + 4 with y so you end up with 2y = y + 12. Again, I think this is easier than the textbook’s way. It also encourages students to understand the substitutions they are making and have more flexible reasoning. I hope you try different substitutions out with your students to find as many different as you can think of. It’s super fun, give it a try!